Emptiness and Universality Problems in Timed Automata with Positive Frequency

Abstract

Abstract. The languages of infinite timed words accepted by timed automata are traditionally defined using Büchi-like conditions. These acceptance conditions focus on the set of locations visited infinitely often along a run, but completely ignore quantitative timing aspects. In this paper we propose a natural quantitative semantics for timed automata based on the so-called frequency, which measures the proportion of time spent in the accepting locations. We study various properties of timed languages accepted with positive frequency, and in particular the emptiness and universality problems.

Frequencies in Forgetful Timed Automata

“…Then, we show that in n-clock forgetful timed automata where time diverges necessarily along a run, the set of frequencies can also be computed thanks to the corner-point abstraction. On the one hand, the result for timed automata for which all cycles are forgetful (strong forgetfulness) is as constructive as the theorem of [BBBS11] over one-clock timed automata. On the other hand, to relax strong forgetfulness and consider timed automata whose simple cycles are forgetful, the proof relies on a set of canonical runs whose frequencies cover the set of all frequencies in the timed automaton.…”

“…Résumé : Une sémantique quantitative pour les mots temporisé infini basée sur la fréquence d'une exécution aété introduite dans [BBBS11]. Malheureusement, la plupart des résultat sont obtenus pour des automates temporisés une seule horloge parce que les techniques ne permettent pas de traiter certains phénomnes de convergences entre horloges.…”

“…Furthermore, the universality problem is proved to be non primitive recursive for non-deterministic timed automata with one clock and undecidable with several clocks. The techniques from [BBBS11] do not extend to timed automata with several clocks, and all counterexamples rely on some phenomenon of convergence between clocks along cycles. Beyond zenoness (when time converges along a run), other convergence phenomena between clocks were first discussed in [CHR02].…”

“…In particular, one can refine the acceptance condition by considering the proportion of time elapsed in accepting locations. A quantitative semantics for infinite timed words based on this notion of frequency has thus been introduced in [BBBS11]. Lower and upper bounds of the set of frequencies of one-clock timed automata are computed using the corner-point abstraction, a refinement of the classical region abstraction, introduced in [BBL08].…”

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Abstract:A quantitative semantics for infinite timed words in timed automata based on the frequency of a run is introduced in [BBBS11]. Unfortunately, most of the results are obtained only for one-clock timed automata because the techniques do not allow to deal with some phenomenon of convergence between clocks.

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